Least-squares linear regression is a statistical technique that may be used to estimate the total cost at the given level of activity (units, labor/machine hours etc.) based on past cost data. It mathematically fits a straight cost line over a scatter-chart of a number of activity and total-cost pairs in such a way that the sum of squares of the vertical distances between the scattered points and the cost line is minimized. The term least-squares regression implies that the ideal fitting of the regression line is achieved by minimizing the sum of squares of the distances between the straight line and all the points on the graph. Assuming that the cost varies along y-axis and activity levels along x-axis, the required cost line may be represented in the form of following equation: y = a + bx In the above equation, a is the y-intercept of the line and it equals the approximate fixed cost at any level of activity. Whereas b is the slope of the line and it equals the average variable cost per unit of activity. Formulas By using mathematical techniques beyond the scope of this article, the following formulas to calculate a and b may be derived: Unit Variable Cost = b = nΣxy - Σx. Σy nΣ x 2 - Σx 2 Total Fixed Cost = a = Σy - bΣx n Where, n is number of pairs of units—total-cost used in the calculation; Σy is the sum of total costs of all data pairs; Σx is the sum of units of all data pairs; Σxy is the sum of the products of cost and units of all data pairs; and Σx 2 is the sum of squares of units of all data pairs.
The following example based on the same data as in tries to illustrate the usage of least squares linear regression method to split a mixed cost into its fixed and variable components: Example Based on the following data of number of units produced and the corresponding total cost, estimate the total cost of producing 4,000 units. Use the least-squares linear regression method.
Coefficients for the Least Squares Regression Line. Insert your data into an Excel spreadsheet. (See Accessing Excel data from the computer lab) Insert a row The first column of the last two rows of the output contain the coefficients of the least-squares reqression line. The first number is the intercept.
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![Squares Squares](http://slideplayer.com/5059601/16/images/15/Estimated+Regression+Equation.jpg)
Least Square Regression Line (LSRL equation) method is the accurate way of finding the 'line of best fit'. Line of best fit is the straight line that is best approximation of the given set of data. It helps in finding the relationship between two variable on a two dimensional plane. It can also be defined as 'In the results of every single equation, the overall solution minimizes the sum of the squares of the errors. Follow the below tutorial to learn least square regression line equation with its definition, formula and example.
X Value Y Value X.Y X.X 60 3.1 60. 3.1 =186 60. 60 = 3600 61 3.6 61. 3.6 = 219.6 61. 61 = 3721 62 3.8 62. 3.8 = 235.6 62.
62 = 3844 63 4 63. 4 = 252 63. 63 = 3969 65 4.1 65.
4.1 = 266.5 65. 65 = 4225 Step 3: Now, Find ∑X, ∑Y, ∑XY, ∑X 2 for the values ∑X = 311 ∑Y = 18.6 ∑XY = 1159.7 ∑X 2 = 19359 Step 4: Substitute the values in the above slope formula given. Slope(b) = (N∑XY - (∑X)(∑Y)) / (N∑X 2 - (∑X) 2) = ((5).(1159.7)-(311).(18.6))/((5).(19359)-(311) 2) = (5798.5 - 5784.6)/(96795 - 96721) = 0.3783292 Step 5: Now, again substitute in the above intercept formula given. Intercept(a) = (∑Y - b(∑X)) / N = (18.6 - 0.3783292(311))/5 = -7.964 Step 6: Then substitute these values in regression equation formula Regression Equation(y) = a + bx = -7.964 + 0.188x Suppose if we want to calculate the approximate y value for the variable x = 64 then, we can substitute the value in the above equation Regression Equation(y) = a + bx = -7.964 + 0.188(64) = 4.068.